Type the correct answer in the box. Use numerals instead of words. If necessary, use / for the fraction bar.

What value of [tex]$x$[/tex] satisfies this equation?
[tex]$1.5(4)^{2x}=12$[/tex]

Round your answer to the nearest hundredth.

The value of [tex][tex]$x$[/tex][/tex] is [tex]$\square$[/tex]



Answer :

To solve the equation [tex]\( 1.5(4)^{2z} = 12 \)[/tex] for [tex]\( z \)[/tex], we can proceed with the following steps:

1. Isolate the exponential term:
Divide both sides of the equation by 1.5 to isolate the term involving [tex]\( z \)[/tex]:
[tex]\[ (4)^{2z} = \frac{12}{1.5} \][/tex]
Simplify the fraction:
[tex]\[ (4)^{2z} = 8 \][/tex]

2. Apply the logarithm:
Take the natural logarithm (logarithm base [tex]\( e \)[/tex]) of both sides to solve for [tex]\( z \)[/tex]:
[tex]\[ \log((4)^{2z}) = \log(8) \][/tex]

3. Use the power rule of logarithms:
The logarithm of an exponentiated term can be simplified using the power rule, which states [tex]\(\log(a^b) = b \log(a)\)[/tex]. Applying this rule, we get:
[tex]\[ 2z \cdot \log(4) = \log(8) \][/tex]

4. Solve for [tex]\( z \)[/tex]:
To solve for [tex]\( z \)[/tex], divide both sides by [tex]\( 2 \log(4) \)[/tex]:
[tex]\[ z = \frac{\log(8)}{2 \log(4)} \][/tex]

5. Evaluate the logarithms:
Consider the values of the logarithms involved and calculate:
[tex]\[ \log(8) \text{ and } \log(4) \][/tex]

6. Compute the fraction:
Calculate the value of [tex]\( z \)[/tex]:
[tex]\[ z \approx 0.75 \][/tex]

7. Round to the nearest hundredth:
Finally, the value of [tex]\( z \)[/tex], when rounded to the nearest hundredth, is 0.75.

So, the value of [tex]\( x \)[/tex] that satisfies the equation is [tex]\( 0.75 \)[/tex].